How to put a fit condition such that the fitted curve is not above the fitted data (Fit<Fitted Data)?

Hello Matlab community,

I am a bit stuggling to understand if there is an easy way to implement the condition above! I am using the fittype function:

aFittype = fittype(expression)

and fitting an easy exponential decay. I tried using a weighting factor, to add a constraint on the fit, but it is not enough.

For physical reason, the exponential decay fitted would make more sense if its values would less than the fitted data. I am then subtracting this as a background.

Thank you for the suggestions!

Answer 1

[Torsten]

In each measurement point, add the constraint

y_measurement >= fun(x_measurement)

where "fun" is your exponential curve.

You will have to use "fmincon" instead of "fit" to incorporate the constraints either in matrix A and vector B or in function "nonlcon".

[Au.]

Hi Torsten, thank you for your answer and sorry for waiting so long before to try your suggestion.

I don't understand much what to do, i am very unfamiliar with all that.

I will give fmincon a vector [x] as input - a set of known values (?);

I will have [fval] as output: that is supposed to be the fitted data that are respecting the conditions I set in fmincon?

(I pretty much know the following doesn't work, but I would like to understand what i am doing ...)

[x,fval]=fmincon(fun,x0,A,B,Aeq,Beq,Lb,Ub,nonlcon)

That is the function i want to fit:

objective = @(x,cte,T) cte*exp(-x/(8.618e-5).*T)
%     xo=[....]; % this is a vector of known values
%     cte=1; % this has to be optimized
%     T=4500; % this has to be optimized
    
A=[];B=[];Aeq=[];Beq=[]; % i have no linear equalities and inequalities
Lb=[0 3000]; % ideally i want to set boundaries on cte and T
Ub=[200 20000]; % this won't work because x is in the function 
nonlcon=@nlcon

That is the nonlinear condition: i should find A and T such that the exponential(x,A,T)-Data<0 (the data has many colunm and each of it should have the condition above...), which i translated:

    function [c,ceq]=nlcon(x,cte,T)
c=objective(x,cte,T)-Data(:,1); 
ceq=[];
    end

I am very excited about making this work (meanwhile i found another way to do work with the data, but this is interesting!)

[Au.]

So finally i got some local help and we implemented something more correct from the use of the function and fitting procedure, but it does not exactly work in the sense that the condition is not fullfilled for some data points- any idea welcome :)

pwd=('...');
cd(pwd)
global x y
x=importdata('xdata.txt');
y=importdata('ydata.txt');

fun = @(p) sum((p(1)*exp(-x/(8.618e-5.*p(2))))-y)^2 % p(1) is amplitude and p(2) temperature
p0=[10,3300];
A=[];B=[];Aeq=[];Beq=[]; % i have no linear equalities and inequalities
Lb=[0 3000]; % ideally i want to set boundaries on cte and T
Ub=[200 20000]; 
nonlcon=@nlcon % non linear condition

p=fmincon(fun,p0,A,B,Aeq,Beq,Lb,Ub,nonlcon) % will return the fitting parameters cte and T

function [c,ceq]=nlcon(p)
global x y
difference=(p(1)*exp(-x/(8.618e-5.*p(2))))-y; %  condition
c = []; % inequality non linear condition
ceq = sum(difference > 0); % equality non linear condition
end

[Torsten]

fun = @(p) sum( ( p(1)*exp(-x/(8.618e-5*p(2))) - y ).^2 )

instead of

fun = @(p) sum((p(1)*exp(-x/(8.618e-5.*p(2))))-y)^2

and

c = difference;
ceq = [];

instead of

c = [];
ceq = sum(difference > 0);

[Au.]

Thanks!